Questions tagged [calculus-of-variations]

This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.

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How to write the Euler-Langrange equation of elliptic PDE containing $\nabla$ like $\Delta u + \nabla u . \theta = f$.

How to write the Euler-Langrange equation of elliptic PDE containing $\nabla$ like $\Delta u + \nabla u . \theta = f$. For some question we can make the $\nabla$ term dissapear by transformation, then ...
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If $c_n \nearrow c$ then $\lim_n \inf_{\pi \in \Pi(\mu, \nu)} \int c_n d\pi = \inf_{\pi \in \Pi(\mu, \nu)} \int c d\pi$

Disclaimer This thread is meant to record. See: SE blog: Answer own Question and MSE meta: Answer own Question. Anyway, it is written as problem. Have fun! :) Let $X,Y$ be Polish spaces and $c:X \...
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Which function $\varepsilon(t)$ maximises the asymptotic growth of this integral?

This question is a continuous version of my earlier (discrete) puzzle - I'm hoping that an answer in the continuous case will shed light on the discrete case. Let $\varepsilon$ be a function $\mathbb{...
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Free boundary geodesics as a critical point of the energy functional

As a consequence of the formula for the first variation of the energy of a curve, we have the following known characterization of geodesics. A piecewise differentiable curve $c:[0,1]\to M$ is a ...
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Gradient of a functional defined on an Hilbert space (with respect to a $W^{1,2}$ inner product)

$\newcommand{\R}{\mathbb R}$ Consider the Hilbert space $X = W^{1,2}(\R)\oplus W^{1,2}(\R)$ (Sobolev spaces). I define a function $F:X\to \R$ as $$F(u,g)= \int_\R u(t)\partial_tg(t) dt.$$ $F$ is ...
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Minimization of a classic functional over $H^1(\Omega)$

Consider the following minimization problem problem $$\underset{w \in H^1(\Omega)}{min} F_\epsilon(w), \quad F_\epsilon(w):= \int_{\Omega} |\nabla w|^2 \, \mathrm{d}x + \frac{1}{\varepsilon^2} \int_{\...
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Showing that a maximum exist for a semi lower sequentially continuous mapping from Hilbert space to R

Let H be a Hilbert space over $R$ , $r > 0$ and $F ∈ C^1(H, R)$ such that: 1)−F is weakly sequentially lower semicontinuous 2) $DF(u) = 0$ implies $u = 0$ (this is the Frechet derivative) 3) $F(0) =...
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How to find the extremum in this variational problem?

I am finding this difficult because this is quite a unique question. In this one, I have to find the Extremum of the functional, $$I(y)=\int_0^1(xy+y^2-2y^2y')dx $$ Boundary conditions, $y(0)=1, ~ y(1)...
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Euler-Lagrange equation, derivative calculation

Let we have $F(u)=\int_A\mathcal{L}(x,u,u')dx$, where $dx$ is Lebesgue measure, $A$ is open and bounded with regular boundary, $\mathcal{L}\in\mathcal{C}^2(A\times\mathbb{R}\times\mathbb{R}^n)$ and ...
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How can I find a function that minimizes this given cost function?

I am trying to find a function $f(x)$ that minimizes the following cost function $$E = \left(\int_{-p}^p{{x^2\mathcal{N}(x)}f(x)}dx-\epsilon\right)^2$$ With $\epsilon\geq 0$ and $\mathcal{N}(x)$ a ...
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How do I find the loop passing through certain points which maximizes a one form integral?

Suppose I have some points in a plane, and I want to integrate a form over a loop passing through all those points, how would I choose out of such loops the one which extremizes the integral? ...
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Prove that $g$ is constant on the interval when it's distributional derivative is zero.

Lemma: Let $g\in \mathcal{C}([0,1])$ be such that $\int_{[0,1]}g(x)\chi'(x)dx=0 \ \forall \chi\in\mathcal{C}_c^\infty(0,1)$ Then $g$ is constant on the interval $[0,1]$. Proof: Let $f\in\mathcal{C}^1(...
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Finding extremal for the functional $J(y)=\int_0^1y'\sqrt{1+(y'')^2}dx,$

For the functional $J$ defined by $$J(y)=\int_0^1y'\sqrt{1+(y'')^2}dx,$$Find an extremal satisfying the conditions $y(0)=0,~y'(0)=0,~y(1)=1$ and $y'(1)=2$? My attempt: Let $F(x,y,y',y'')=y'\sqrt{1+(y''...
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Why does the arclength functional take same form in rotated coordinate systems?

I'm going through the textbook "Emmy Noether's Wonderful Theorem" by Dwight Neuenschwander. Therein, the author defines the coordinate transformation (infinitesmal rotation of orthogonal ...
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Convergence in L^p for function multiplied for another function

Problem: Let $\Omega \subset R^n$ and let $f_{\epsilon}, f \in W^{1,p}(\Omega)$ be functions such that $f_{\epsilon} \rightarrow f$ in $L^p(\Omega)$. Which condition should have another function $g$ ...
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Can we apply fundamental lemma of calculus of variations

Let us assume that $t_0$ and $t_1$ are some constants, and $f(t)$ and $u(t)$ is a continuous mapping from $[t_0,t_1]$ to $\mathbb{R}^p$ and $\mathbb{R}^m$, respectively. Suppose that $$\int_{t_0}^{t_1}...
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Extremals of $\int_0^\pi(y')^2dx$

I am trying to show that $\int_0^\pi(y')^2dx$ with $y(0) = 0$, $y(\pi) = 0$ has infinitely many extremals subject to the constraint $\int_0^\pi y^2 dx = \pi/2$. I know that the first-order necessary ...
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Can we characterize functions, like exponentials, the gamma function, and tetration, as solutions of an optimization problem?

This is something I recently started wondering about. I've long been interested in the idea of problems of the form "given a sequence of real numbers $a_n$, under what cases is there some way to ...
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Texts about global minima in functionals

I am doing some numerics where I found the minimal value of a functional that does not satisfy the Euler-Lagrange equation associated. I think I am dealing with a minimal value that is not a local ...
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Variational Derivative Calculation

I am quite new to this concept of variational derivatives and am struggling to figure out how one would compute such a thing in practice. The example I am looking at is for: $$ J[c] = \int_{\partial \...
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Constrained variational calculus: find extremums of $\int_0^\infty ay(t)^2 + by'(t)^2 + f(t)y(t) \ \text{d}t$ subject to $0 \leq y(t)\leq k$

I wish to find extremums of a functional $J[y]$ that is given by $$ J[y] =\int_0^\infty ay(t)^2 + by'(t)^2 + f(t)y(t) \ \text{d}t \hspace{0.5cm} \text{subject to} \hspace{0.5cm} 0 \leq y(t)\leq k $$ ...
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Equivalent functional integral of ODEs

I was going through the Ritz method in Finite Element analysis when I came across the statement here (see equation 1.25, its not same but similar. below I have taken a problem from my book.), ...
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Practical Question - How to Optimise Times over a Timeline

Let's assume you want to buy a blueberry muffin at your local bakery. But, there are a few things that complicate the matter: The bakery is open from 6am - 1pm. Blueberry muffins are in high demand ...
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Variational form of 2nd order linear ODE

I have been having issues getting the variational form of the following differential equation. $$ \frac{d^2u}{dx^2} - u = -1 $$ I looked to multiply by $u$ and integrate over the length for the ...
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Maximal entropy function of multiple random variables

Assume that we have a known multivariate distribution of $N$ variables denoted by $f(\vec{X})$. Consider the random variable $Y = g(\vec{X})$ defined by a smooth function $g:\mathbb{R^{n}}\rightarrow\...
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Dependence of energy of solution of IVP on the initial condition

Let $f$ be a smooth function $[0, L] \times \mathbb{R} \to \mathbb{R}$, and consider an initial value problem (IVP) of the form $$ \begin{cases} x' = f(t, x),\\ x(0) = x_{0} \in [0, 2\pi). \end{cases}...
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Formal definition of variation

I'm trying to read a book on classical mechanics, and I'm having a hard time trying to know what exactly is a variation. In the Lagrangian "formalism" the differential forms $dx$ are changed ...
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Sum of Squares Optimization of an Numerical Integral

I am trying to evaluate the following minimize-sum-square optimization expression: $$\min_{a\in A}\sum_{x\in X}(\int_{-\infty}^{\infty} e^{-itx}\phi(t, a) dt - \int_{-\infty}^{\infty} e^{-itx}\phi(t, ...
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Would second order variation necessary for the calculation of the area?

In physics people do the variation of a quantity typically of the form $$ \delta_X f(X,Y,Z)=0 $$ where in general the variation $$X\rightarrow X+\delta X $$ where $$\delta X\propto \epsilon g(X) $$ of ...
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Blow up of the function $G$ implies blow up of its integral

Let $\Omega \subset \mathbb{R}^N$ be a bounded domain, $g \in C(\overline{\Omega} \times \mathbb{R}, \mathbb{R})$ and $g$ bounded. If $G(x,u) := \int_0^u g(x,s) ds$, show that $$\lim\limits_{|s| \...
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Along what curve does a circle roll down the fastest?

In this MO question, I stated that a circle rolls down the fastest along a cycloid curve - as shown by Johan Bernoulli, thereby solving the Brachistochrone problem. However, as user Manfred Weis ...
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Euler's equation different boundary conditions

I am trying to find the extremal of a functional I, the boundary conditions on the integral are [0,π/6] but the boundary conditions are y(0) = 0 and y(π) = 2, I am confused about why the boundary ...
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Finding the path of a light ray using differential geometry

Hi I am trying to solve a calculus of variations problem I would like to solve it using differential geometric approach but i am not sure. As an example how would I go back doing it for the example ...
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Why we need a Hausdorff measure here?

If $\varphi\in \mathcal{C}^1_c(A)$ and $G\in \mathcal{C}^1(A,\mathbb{R}^n)$, then $\int_A \text{div} (\varphi G(x))\ dx=0$. ($K$ is compact support) Proof: $I=\int_B \text{div} (\varphi \ G(x))\ dx=\...
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Proving that certain functional has a minimum

Let $I$ be a closed interval, and let $y$ be a smooth function $I \to (0, \pi)$. Consider the functional $$ J(y) = \int_{U} \frac{\left(F(x,y(x))^{2}+G(x,y'(x))^{2}\right)^{2}}{F(x,y(x))^{2}}\, dx, $$ ...
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what to make out of it when Euler-Lagrange Equation is constantly zero

I'm new to Calculus of Variations, and I'm trying to apply it to a simple vector calculus problem. Let's consider finding a curve $C$ along which the work $W$ done by a given vector field $\textbf{F}$ ...
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Euler-Lagrange equation w.r.t. compact variation

I follow the lecture of Calculus of variations, the topic is Euler-Lagrange's equations. $$ \mathcal{L}: \ A\times\mathbb{R}\times\mathbb{R}^n\to \mathbb{R}, \ x\in A, \ u\in \mathbb{R}, \ \xi\in \...
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2 votes
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Proving that certain functional has a positive lower bound

Let $I =[a,b]$, and let $\theta$ denote a smooth function $I \to (0, \pi)$. Consider the functional $$ J(\theta) = \int_{a}^{b} \frac{\left(F(s,\theta(s))^{2}+G(s,\theta'(s))^{2}\right)^{2}}{F(s,\...
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A variational argument involving the Rayleigh quotient from Gilbarg-Trudinger

I am currently reading Gilbarg-Trudinger's discussion of the eigenvalues of a self-adjoint scond-order elliptic operator. Let $Lu = D_i(a^{ij}D_ju + b^iu) - b^iD_iu + cu$ with $a^{ij}$ symmetric. ...
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Why exactly do we write $\mathcal{L}(x(t),x'(t),t)$ instead of simply $\mathcal{L}(x(t),t)$?

I have many times seen Lagrangian written as $\mathcal{L}(x(t),x'(t),t)$. I undestand that this is a function of $x(t)$, $x'(t)$ and $t$. So it can theoretically look something like $$\mathcal{L}(x(t),...
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Can the geodesic equation be used to solve the Brachistochrone Problem?

Assume the initial condition is that a point mass starts at height $y_0$. After descending to height $y < y_0$, we know that its speed will be $v = \sqrt{2mg(y_0 - y)}$. Thus, the displacement ...
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confusion about the meaning of a stationary integral

In calculus if variation, There are problems where I have to find the function (curve) that makes the value of an integral minimum between two points. It has to be stationary, in a sense that an ...
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Integral equality in a matrix setting

While reading a proof I stumbled over the following equation: \begin{align} \int_{B_1} \langle A,D\psi(x)\rangle dx=0 \quad \forall \psi\in W^{1,\infty}_0(B_1,\mathbb{R}^2)\forall A\in \mathbb{R}^{2\...
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Solving $- \Delta u = \lambda u + |u|^{q-2} u$ by method of Langrange's multipliers

Let $\lambda < \lambda_1$ and $ 2 < q < 2^*$. I am trying solve this problem $$(P) \begin{align} \begin{cases} -\Delta u &= \lambda u + |u|^{q-2} u \ \text{in} \ \Omega,\\ u &= 0 \ \...
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Finding the path between to points which minimizes cost function subject to constraints

I have a question regarding calculus of variations. Given a cost function in configuration space how can I find the path between two points which minimizes it subject to constraints? I am aware that ...
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5 votes
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Problem 9 M.L Krasnov variational calculus

Warning: Finding extreme value of a multivariable function My question differs from this since I try to use the Hessian criterion so it is not a repeated question. My question: Problem 9 M.L Krasnov. ...
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How to solve this problem with calculus of variation?

Let $c \in (0,1)$ and $n\geq 2$ be some integer. Suppose we can choose twice differentiable function $g: [0,1]\to R$ to solve the following inequality constrained program $$max_g \int_{c }^1 x g(x)dx$...
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What $is$ $\text{dist}^2\left( F,\mathsf{SO}(3)\right)$?

Reading some review about calculous of variations and mechanics, the following notation is very recurrent for energy density functionals \begin{equation} \text{dist}^p\left( F,\mathsf{SO}(N)\right), \...
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Inverse function in an Euler-Lagrange Equation

I'm interested in minimizing the functional $I[f]=\int_{a}^{b} \mathcal{L}(x,f,f',f'',f^{-1},(f^{-1})', (f^{-1})'') dx$. Would it be allowable to consider that $f^{-1}$ is some function $g$ and apply ...
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Capturing $\infty$ endpoint and additional boundary conditions

I'm interested in minimizing the functional $I[f]=\int_{a}^{\infty} J(x,f(x),f'(x)) dx$ but the boundary conditions that I have on $f$ are a bit "weird": one endpoint is fixed in the ...
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